Can we ever achieve hypercomputation using recurrent neural networks?
I presume the proof the OP is referring to can be found in this monograph by Hava Siegelmann?

In his article 'The Myth of Hypercomputation', the eminent computer scientist Martin Davis explains (p8-9) that there is nothing 'super Turing' about this formulation.

EDIT: It's looking like the claim about rational weights being super-Turing is made in this  more recent paper by Siegelmann, which introduces an additional assumption of plasticity, i.e. that weights can be dynamically updated.